收到退货（礼物）后，我应该把钱退给下订单的买家？还是收件人？

谁付的钱，你就退给谁啊。不要管这个收件人。他发信息可能是想确保钱能退给买家。
钱只要退给下订单的人就好。如果这个收件人有问题，那可以自己向亚马逊反映

看链接：
Return a Gift
https://www.amazon.com/gp/help/customer/display.html?nodeId=201077700 
里面有说：”If your gift return is authorized, Amazon or the Seller will process your refund as an Amazon.com Gift Card after your return is received.
If you're the gift giver, your refund will be processed to the original payment method.”
退货后，收礼物的人会得到亚马逊Gift Card；而送礼物的人会拿到退款。所以把钱退给交易的另一方就好。